Final answer:
To find the equation of the tangent line to the graph of the function f(x) = x + 1/x at the point x = 2, we need to find the slope of the tangent line at that point and then substitute the coordinates of the point into the formula. The slope of the tangent line at x = 2 is 3/4. The equation of the tangent line is y - 5/2 = (3/4)(x - 2).
Step-by-step explanation:
The equation of a tangent line to a graph is given by the formula: y - y1 = m(x - x1), where (x1, y1) is a point on the graph and m is the slope of the tangent line at that point. To find the equation of the tangent line to the graph of the function f(x) = x + 1/x at the point x = 2, we need to find the slope of the tangent line at that point and then substitute the coordinates of the point into the formula.
To find the slope of the tangent line at x = 2, we can take the derivative of the function f(x) = x + 1/x and evaluate it at x = 2. The derivative of f(x) is found using the quotient rule:
f'(x) = (1 - 1/x^2).
Substituting x = 2 into f'(x), we get:
f'(2) = (1 - 1/2^2) = (1 - 1/4) = 3/4.
Therefore, the slope of the tangent line at x = 2 is 3/4. Now, we substitute the coordinates (2, f(2)) = (2, 2 + 1/2) = (2, 5/2) into the equation of the tangent line:
y - 5/2 = (3/4)(x - 2).
This is the equation of the tangent line to the graph of the function f(x) = x + 1/x at the point x = 2.